EQUILIBRIUM OF A PARTICLE 29 is perpendicular to the string. Show that 21 = jua, where I is the length of the string, a that of the rod, and fj, the coefficient of friction. 16. A particle resting upon an inclined plane is at the point of motion under the action of the force F, which acts downward along the plane. If the angle of elevation of the plane is changed from o^ to a% and the direction of the force reversed the particle will barely start to move up the plane. Express //, in terms of cti and a>>. 17. A string, which passes over the vertex of a rough double inclined plane, supports two weights. Show that the plane must be tilted through an angle equal to twice the angle of friction, in order to bring it from the position at which the particles will begin to move in one direction to the position at which they will begin to move in the opposite direction. 18. Three equal spheres are placed on a smooth horizontal plane and are kept together by a string, which surrounds them in the plane of their centers. If a fourth equal sphere is placed on top of these, prove that the W tensile force in the string is ——z., where W is the weight of each sphere. OvD 19. Three equal hemispheres rest with their bases upon a rough horizontal plane and are in contact with one another. What is the least value of JJL which will enable them to support a smooth sphere of the same radius and material? 20. If the center of gravity of a rod is at a distance a from one end and b from the other, find the least value of n which will allow it to rest in all positions upon a rough horizontal ground and against a rough vertical wall. 21. A string, which is slung over two smooth pegs at the same level, supports two bodies of equal weight W at the ends, and a weight W at the middle by means of a smooth ring through which it passes. Find the position of equilibrium of the middle weight.